The number of distinct terms in the expansion of `(x+y^(2))^(13)+(x^(2)+y)^(14)` is (a) 27 (b) 29 (c) 28 (d) 25

Options

A. `27`
B. `29`
C. `28`
D. `25`

Explanation

To solve the problem of finding the number of distinct terms in the expansion of \((x+y^2)^{13} + (x^2+y)^{14}\), we will follow these steps:

### Step 1: Determine the number of terms in each expansion
1. **Expansion of \((x+y^2)^{13}\)**:
- The number of distinct terms in the expansion of \((a+b)^n\) is given by \(n + 1\).
- Here, \(n = 13\).
- Therefore, the number of distinct terms is \(13 + 1 = 14\).

2. **Expansion of \((x^2+y)^{14}\)**:
- Similarly, for \((x^2+y)^{14}\), \(n = 14\).
- The number of distinct terms is \(14 + 1 = 15\).

### Step 2: Calculate total terms before considering common terms
- The total number of terms from both expansions is:
\[ 14 + 15 = 29 \]

### Step 3: Identify common terms
- To find the common terms, we need to equate the powers of \(x\) and \(y\) from both expansions.

1. **From \((x+y^2)^{13}\)**:
- A general term can be expressed as:
\[ T_1 = C(13, r_1) x^{13 - r_1} (y^2)^{r_1} = C(13, r_1) x^{13 - r_1} y^{2r_1} \]
- Here, the power of \(x\) is \(13 - r_1\) and the power of \(y\) is \(2r_1\).

2. **From \((x^2+y)^{14}\)**:
- A general term can be expressed as:
\[ T_2 = C(14, r_2) (x^2)^{r_2} y^{14 - r_2} = C(14, r_2) x^{2r_2} y^{14 - r_2} \]
- Here, the power of \(x\) is \(2r_2\) and the power of \(y\) is \(14 - r_2\).

### Step 4: Set up equations for common terms
- For the terms to be common, we need:
\[ 13 - r_1 = 2r_2 \quad \text{(1)} \]
\[ 2r_1 = 14 - r_2 \quad \text{(2)} \]

### Step 5: Solve the equations
1. From equation (1):
\[ r_2 = \frac{13 - r_1}{2} \]
2. Substitute \(r_2\) in equation (2):
\[ 2r_1 = 14 - \frac{13 - r_1}{2} \]
Multiply through by 2 to eliminate the fraction:
\[ 4r_1 = 28 - (13 - r_1) \]
Simplifying gives:
\[ 4r_1 = 28 - 13 + r_1 \]
\[ 4r_1 - r_1 = 15 \]
\[ 3r_1 = 15 \implies r_1 = 5 \]
3. Substitute \(r_1\) back to find \(r_2\):
\[ r_2 = \frac{13 - 5}{2} = \frac{8}{2} = 4 \]

### Step 6: Conclusion on common terms
- We found one common term corresponding to \(r_1 = 5\) and \(r_2 = 4\).

### Step 7: Calculate distinct terms
- The number of distinct terms is:
\[ \text{Total terms} - \text{Common terms} = 29 - 1 = 28 \]

### Final Answer
The number of distinct terms in the expansion is **28**.